Question: Create a rational function f(x) such that the function has:
2 vertical asymptotes
1 hole
1 positive horizontal asymptote
2 vertical asymptotes
y = 1/(x^2-1)
a hole
y = (x+2) / (x+2)(x^2-1)
a horizontal asymptote. y > 0
needs top and bottom to be of the same degree
y = (x+2)^3 / (x+2)(x^2-1)
asymptote is y=1
To create a rational function with the specified criteria, we can use the following steps:
Step 1: Start with a basic rational function of the form f(x) = (ax^m + bx^n + c)/(dx^p + ex^q + f), where m, n, p, q are integers and a, b, c, d, e, f are constants.
Step 2: Determine the values for m, n, p, q based on the given requirements.
To have 2 vertical asymptotes, we need the denominator to have at least two factors that cannot be cancelled out by factors in the numerator. We can achieve this by setting p = 1 and q = 1.
To have 1 hole, we can set one of the factors in the numerator to cancel out one of the factors in the denominator. Let's set m = p, and n = q, so that the hole will occur at x = 0.
To have 1 positive horizontal asymptote, we need the degree of the numerator to be less than or equal to the degree of the denominator. To ensure this, we can set m = p - 1 and n = q - 1.
Step 3: Substitute the values obtained in step 2 into the basic rational function from step 1.
f(x) = (ax^(p-1) + bx^(q-1) + c)/(dx + ex + f)
This function will have 2 vertical asymptotes, 1 hole at x = 0, and 1 positive horizontal asymptote.
To create a rational function with these characteristics, we'll start by understanding what these concepts mean:
1. Vertical asymptotes: These are vertical lines where the function approaches either positive or negative infinity. A vertical asymptote can occur at any x-value where the denominator of the rational function becomes zero.
2. Hole: A hole in a rational function occurs when both the numerator and denominator have a common factor that can be canceled out. This creates a point (x, y) where the function is not defined, but can be filled in by simplifying.
3. Horizontal asymptote: This is a horizontal line that the function approaches as x approaches positive or negative infinity. For a rational function, the degree of the numerator must be less than or equal to the degree of the denominator for a horizontal asymptote to exist. The location of the horizontal asymptote depends on the ratio of the coefficients of the highest degree terms in the numerator and denominator.
With these concepts in mind, we can create a rational function that meets the given conditions:
Step 1: Choose the denominator
Let's start with two different linear factors in the denominator to create two vertical asymptotes. For example, let's use (x + 1)(x - 3) as the denominator.
Step 2: Choose the numerator
We want a hole to occur in the rational function. To create a hole, we need a common factor in the numerator and denominator. Let's say the common factor is (x + 2).
Step 3: Add a horizontal asymptote
Next, we want a positive horizontal asymptote. This means that the degree of the numerator should be less than or equal to the degree of the denominator. Let's choose the numerator to be a constant, say 5.
Putting it all together, the rational function f(x) that satisfies the given conditions is:
f(x) = (5(x + 2))/((x + 1)(x - 3))
This function has two vertical asymptotes at x = -1 and x = 3, a hole at x = -2, and a positive horizontal asymptote at y = 0.